Use the given functions to find f(g(x)), and give the restrictions on x.
f(x) = 1 / (x-4)
g(x) = (4/x) + 4
What are the restrictions of x? How do you look for that?
f(g(x))
= f(4/x + 4)
= 1/(4/x+4 - 4)
= 1/(4/x)
= x/4
there is no restriction on this final result, BUT
in the original g(x) = 4/x + 4, there was the restriction of x ≠ 0
in my view that restriction should carry over to the final function.
eg
g(3) = 4/3 + 4 = 16/3
f(16/3) = 1/(16/3 - 4) = 3/4
and according to my result f(g(3)) = 3/4, the correct result
g(0) is undefined
f(undefined) = 1/(undefined - 4), but my final function would give us 0
extra credit
f(x) = 1/(x-4) is undefined at x = 4
g(x) = 4/x + 4 is undefined at x = 0
so, why is f(g(x)) not undefined at x=4?
Yup, oobleck is right
so, would be have x ≠ 0,4 ??
So both 0 and 4 would be restrictions?
4 coming from the f(x) and
0 coming from the g(x)
note that g(x) is never equal to 4
To find the restrictions on x, we need to identify any values of x that would result in an undefined output for the given functions. In this case, we have two functions: f(x) = 1 / (x-4) and g(x) = (4/x) + 4.
For f(x), the denominator cannot be equal to zero, as division by zero is undefined. Thus, we need to find the value(s) of x that make (x-4) equal to zero. Solving the equation x-4 = 0, we find that x = 4. Therefore, x cannot be equal to 4 for the function f(x) to be defined.
For g(x), the denominator cannot be equal to zero either. Hence, we must solve the equation x = 0 to find if any restrictions exist. However, looking at the given function g(x) = (4/x) + 4, we can see that the denominator is x. Since the denominator x is in the form of 4/x, there is no restriction on x, even if x equals 0. This is because when x is 0, the expression becomes (4/0) + 4, which is undefined. However, since the original expression for g(x) excludes the value x = 0, there are no restrictions on x for g(x).
Now, to find f(g(x)), we substitute the expression for g(x) into f(x):
f(g(x)) = f((4/x) + 4)
Let's perform the substitution:
f(g(x)) = 1 / (((4/x) + 4) - 4)
Now, simplify the expression to obtain the final answer for f(g(x)).